If R, C and L are fundamental quantities in a circuit like resistance, capacitance and inductance in W, then find dimensional formula for resistance and capacitance.

Dimensional analysis can be defined as the practice of studying relations between physical quantities by recognising the dimensions of them. These dimensions are independent of numerical multiples and similar constants. The quantities present in the world can be described as a function of the fundamental dimensions.

Dimensional Formula

Dimensional formula of a quantity is the expression indicating the powers wherein the fundamental units are to be increased to obtain one unit of a derived quantity. For instance, if Q is an unit of a derived quantity which is defined by Q = M^aL^bT^c, then M^aL^bT^c is known as the dimensional formula with exponents a, b and, c that are called the dimensions.

Dimensional Formula for Resistance

The dimensional formula of resistance can be expressed as:

M¹L²T^-3I^-2

Here,

M = Mass
I = Current
L = Length
T = Time

Resistance (R) = Voltage × Current^-1… [1]

Now, because, Voltage (V) = Electric Field × Distance = [Force × Charge^-1] × Distance

Accordingly, dimensional formula of charge = current × time = I¹T¹

Dimensional formula of voltage = [Force × Charge^-1] × Distance

= [M¹ L¹ T^-2] × [I¹T¹]^-1 × [L¹]

= [M¹ L² T^-3I^-1] … [2]

On replacing equation (ii) in equation (i) we get,

Resistance (R) = Voltage × Current^-1

R = [M¹ L² T^-3I^-1] × [I]^-1 = [M¹ L² T^-3I^-2]

Hence, resistance is dimensionally depicted as M L² T^-3I^-2.

Dimensional Formula of Capacitance

The dimensional formula of Capacitance is expressed by:

M^-1L^-2T⁴I²

Here,

M = Mass
I = Current
L = Length
T = Time

Derivation of Capacitance

Capacitance (C) = Charge × Voltage^-1. . . [1]

Since, Charge = Current × Time

∴ Therefore, the dimensional formula of charge can be give as = [I¹T¹] . . . . [2]

And, Voltage = Electric Field × Distance . . [3]

Now, as per Electric Field,

Electric Field = [Force × Charge^-1]

Hence, dimensional formula of force and charge equals [M¹ L¹ T^-2] and [I¹T¹] respectively.

∴ The dimensional formula of Electric Field = [M¹ L¹ T^-2] × [I¹T¹]^-1

= [M¹ L¹ T^-3 I^-1] . . . [4]

On replacing equation [4] in equation [3] we can obtain,

The dimensional formula of Voltage = [M¹ L¹ T^-3 I^-1] × [L¹]

= [M¹ L² T^-3I^-1] . . . [5]

On replacing equations [5] and [2] in equation [1] we can obtain,

Hence, Capacitance = Charge × Voltage^-1

Or, as can be expressed, C = [I¹T¹] × [M¹ L² T^-3I^-1]^-1 = [M^-1 L^-2 T⁴I²]

Thus, the Capacitance can be dimensionally expressed as [M^-1 L^-2 T⁴I²].

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If R, C and L are fundamental quantities in a circuit like resistance, capacitance and inductance in W, then find dimensional formula for resistance and capacitance.

Dimensional Formula

Dimensional Formula for Resistance

Dimensional Formula of Capacitance

Derivation of Capacitance

Related Questions

CBSE CLASS XII Related Questions

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Write any two features of nuclear forces.

3.
If both the number of protons and the neutrons are conserved in each nuclear reaction, in what way is mass converted into energy (or vice versa) in a nuclear reaction? Explain.

4.
Two heaters rated as \((P_1,V)\) and \((P_2,V)\) are connected in series across a dc source of \(V/2\) volt. The power consumed by the combination will be –

5.
Assertion (A) : All atoms have a net magnetic moment. Reason (R) : A current loop does not always behave as a magnetic dipole.

6.
Photoemission of electrons occurs from a metal (\( \phi_0 = 1.96 \, \text{eV} \)) when light of frequency \( 6.4 \times 10^{14} \, \text{Hz} \) is incident on it. Calculate: Energy of a photon in the incident light, The maximum kinetic energy of the emitted electrons, and The stopping potential.

Similar Physics Concepts

Comments

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If R, C and L are fundamental quantities in a circuit like resistance, capacitance and inductance in W, then find dimensional formula for resistance and capacitance.

Dimensional Formula

Dimensional Formula for Resistance

Dimensional Formula of Capacitance

Derivation of Capacitance

Related Questions

CBSE CLASS XII Related Questions

1. Write any two features of nuclear forces.

3. If both the number of protons and the neutrons are conserved in each nuclear reaction, in what way is mass converted into energy (or vice versa) in a nuclear reaction? Explain.

4.Two heaters rated as \((P_1,V)\) and \((P_2,V)\) are connected in series across a dc source of \(V/2\) volt. The power consumed by the combination will be –

5.Assertion (A) : All atoms have a net magnetic moment. Reason (R) : A current loop does not always behave as a magnetic dipole.

6.Photoemission of electrons occurs from a metal (\( \phi_0 = 1.96 \, \text{eV} \)) when light of frequency \( 6.4 \times 10^{14} \, \text{Hz} \) is incident on it. Calculate: Energy of a photon in the incident light, The maximum kinetic energy of the emitted electrons, and The stopping potential.

Similar Physics Concepts

Comments

SUBSCRIBE TO OUR NEWS LETTER

1.
Write any two features of nuclear forces.

3.
If both the number of protons and the neutrons are conserved in each nuclear reaction, in what way is mass converted into energy (or vice versa) in a nuclear reaction? Explain.

4.
Two heaters rated as \((P_1,V)\) and \((P_2,V)\) are connected in series across a dc source of \(V/2\) volt. The power consumed by the combination will be –

5.
Assertion (A) : All atoms have a net magnetic moment. Reason (R) : A current loop does not always behave as a magnetic dipole.

6.
Photoemission of electrons occurs from a metal (\( \phi_0 = 1.96 \, \text{eV} \)) when light of frequency \( 6.4 \times 10^{14} \, \text{Hz} \) is incident on it. Calculate: Energy of a photon in the incident light, The maximum kinetic energy of the emitted electrons, and The stopping potential.